Browder’s theorem with general parameter space

It follows from Browder (Summa Bras Math 4:183–191, 1960) that for every continuous function $$F : (X \times Y) \rightarrow Y$$ , where X is the unit interval and Y a nonempty, convex, compact subset of locally convex linear vector space, set fixed points F, defined by $$C_F := \{ (x,y) \in :F(x,y)=y\}$$ has connected component whose projection to first coordinate X. We extend Browder’s result case Hausdorff space.